What is the relationship between sets and singleton sets?

[aliendoom]

My book has a description in the setting of a coin flip experiment. It says, if we let "heads" equal 1 and "tails" equal 0, then we get a random variable:

$X=X(\omega)\epsilon\{0,1\}$

where $\omega$ belongs to the outcome space $\Omega=\{heads, tails\}$

Under the innocuous subheading:

"Which are the most likely $X(\omega)$, what are they concentrated around, what are their spread?

the book says that to approach those problems, one first collects "good" subsets of $\Omega$ in a class F, where F is a $\sigma$-field. Such a class is supposed to contain all interesting events. Certainly, {w:X(w)=0}={tail} and {w:X(w)=1}={head} must belong to F, but also the union, difference, and intersection of any events in F and its complement the empty set. If A is an element of F, so is it's complement, and if A,B are elements of F, so are A intersection B, A union B, A union B complement, B union A complement, A intersection B complement, B intersection A complement, etc.

Whaaa? What's all that $\sigma$-field stuff got to do with the probabilites of X(w)? Also, if A and B are a member of a class F, isn't A union B also automatically a member of the class F, as well as A intersection B, etc.?

 

[Hurkyl]

In the general case, taking your event space to be all possible subsets of the outcome space is just too many events to be interesting, so you need to restrict your attention to a sigma-algebra of events upon which you can say interesting stuff.

Of course, this is a nice, simple, well-behaved case, and it is safe to take as your event space all possible subsets of the outcome space.

 

[aliendoom]

In the general case, taking your event space to be all possible subsets of the outcome space is just too many events to be interesting.

I don't understand your use of the phrase "all possible subsets". Why doesn't the event space(I'm assuming that is what F is) contain "all possible outcomes"? What do "all the possible subsets" of "all the possible outcomes" have to do with anything? Furthermore, why should I be concerned with anything other than the outcome space $\Omega$?

In addition, I guess I'm not grasping what a subset is. If A,B,C are in the event space, aren't all subsets of A,B,C automatically in the event space?

 

[Hurkyl]

Why doesn't the event space(I'm assuming that is what F is) contain "all possible outcomes"?

It does -- two of the events are {heads} and {tails}. There are two other events: {heads, tails} (which happens with probability 1), and {} (which happens with probability 0).

This might make more sense if you consider a bigger outcome space: some of the possible events that you could measure about rolling a pair of dice are:
Getting a 7.
Getting any number.
Getting an even number.
Getting doubles.

 

[aliendoom]

This might make more sense if you consider a bigger outcome space: some of the possible events that you could measure about rolling a pair of dice are:
Getting a 7.
Getting any number.
Getting an even number.
Getting doubles

Ok, so the outcome space is:
{

{1,1}, {1, 2}, ...{1, 6}
{2,1}, {2, 2}...{2, 6}
...
...
{6,1}, {6,2}...{6,6}

}
?

Getting a 7 consists of the events/subsets {1,6},{6,1},{2,5},{5,2},{4,3},{3,4}
?

Getting any number consists of the the whole outcome space:
{1,1}, {1, 2}, ...{1, 6}
{2,1}, {2, 2}...{2, 6}
...
...
{6,1}, {6,2}...{6,6}

?

Getting an even number consists of the events/subsets:
{1,1}, {2, 2}, {3,1},{1,3},...etc.
?

Getting doubles consists of the events/subsets:
{1,1}, {2,2}...{6,6}
?

Now what is a sigma field for rolling two dice, and why is it relevant?

 

[Hurkyl]

Now what is a sigma field for rolling two dice, and why is it relevant?

Before we can speak about probability, we must say which events we will consider. In the case of two dice, we've now identified several interesting events, such as rolling doubles.

A "sigma field" encapsulates the properties that we would like events to have. For example, if we consider the events "A" and "B", we would probably also want to consider the event "A or B". This corresponds to the fact that any sigma field is closed under unions.

Now, while this example is more interesting than flipping a coin, it is still pretty boring, since you would probably want to say that any possible subset of the outcomes constitutes an event that we can consider. And that's fine.

This idea becomes much more important, however, when you progress to an infinite space of outcomes, since they have "too many" subsets. For example, if we take the uniform probability measure on the interval [0, 1], there is simply no reasonable way to assign probabilities to some particularly ill-behaved subsets of that interval. So, we have to adopt a sigma-field of events out of necessity.

Another benefit of considering sigma-fields of events is that sometimes we don't care about the difference between some outcomes. For example, sometimes when rolling two dice, we might only want to consider events for which we don't distingush between the two dice. In such a case, we might consider a sigma-field that contains only symmetric events, like {(1, 1), (1, 2), (2, 1)}, and not asymmetric events like {(1, 2)}.

 

[aliendoom]

Hi,

Thanks for the response.

This idea becomes much more important, however, when you progress to an infinite space of outcomes, since they have "too many" subsets. For example, if we take the uniform probability measure on the interval [0, 1], there is simply no reasonable way to assign probabilities to some particularly ill-behaved subsets of that interval. So, we have to adopt a sigma-field of events out of necessity.

What is a uniform probability measure? Do you mean that any real number in the interval [0,1] has an equal probability of occurring? What would be an example of an ill behaved subset of that interval?

After re-reading this whole thread several times, it seems to me that a sigma field is some sort of conditional probability, i.e. given that an event is in your sigma field, then you can talk about the probability of the event. Is that right?

I also gather that this is incorrect:

Also, if A and B are a member of a class F, isn't A union B also automatically a member of the class F, as well as A intersection B, etc.?

In other words if you have a set like this:

{A, B}

that is different from a set like this:

{A, B, {A,B} }

Is this different too:

{ {A}, {B}, {A,B} }

and how so?

 

[Hurkyl]

The defining property of {A} is that it contains the object A and nothing else. More formally:

$$x \in \{A\} \Leftrightarrow x = A$$

However, note that we cannot say that $A \in A$. In fact, A might not even be a set, so it wouldn't even make sense to speak of A having elements! Therefore A and {A} cannot be the same thing.


There is, of course, a close relationship between the class of objects and the class of singleton sets -- I can turn any object into a singleton set by "putting braces around it", and I can turn any singleton set into an object by "taking the braces off".